Measuring the dimension of partially embedded networks

Scaling phenomena have been intensively studied during the past decade in the context of complex networks. As part of these works, recently novel methods have appeared to measure the dimension of abstract and spatially embedded networks. In this paper we propose a new dimension measurement method for networks, which does not require global knowledge on the embedding of the nodes, instead it exploits link-wise information (link lengths, link delays or other physical quantities). Our method can be regarded as a generalization of the spectral dimension, that grasps the network's large-scale structure through local observations made by a random walker while traversing the links. We apply the presented method to synthetic and real-world networks, including road maps, the Internet infrastructure and the Gowalla geosocial network. We analyze the theoretically and empirically designated case when the length distribution of the links has the form P(r) ~ 1/r. We show that while previous dimension concepts are not applicable in this case, the new dimension measure still exhibits scaling with two distinct scaling regimes. Our observations suggest that the link length distribution is not sufficient in itself to entirely control the dimensionality of complex networks, and we show that the proposed measure provides information that complements other known measures

Techniques for better vario-scale map content

The previous chapters covered the research where the vario-scale structure has been introduced. The main aim of the research was general functionality, performance and optimization. So far, the technical aspects had higher priority than the map content. Therefore, this chapter focuses on improving our development kit for generating varioscale content. It presents a strategy to provide good cartographic results throughout all scales and properly stored in the structure. First, Section 4.1 specifies our target. Section 4.2 presents solutions of other researchers. Section 4.3 introduces concepts and tools which are used later in newly designed process. This is demonstrated on road features in Section 4.4. The section presents the generalization approach for the whole scale range from large scale, where roads are represented as area objects, to mid and small scales, where roads are represented as line objects. In our suggested gradual approach even for one road the representation can be mixed, both area and line, which may provide better transition phase and b

Design and development of a system for vario-scale maps

Nowadays, there are many geo-information data sources available such as maps on the Internet, in-car navigation devices and mobile apps. All datasets used in these applications are the same in principle, and face the same issues, namely: Maps of different scales are stored separately. With many separate fixed levels, a lot of information is the same, but still needs to be included, which leads to duplication. With many redundant data throughout the scales, features are represented again and again, which may lead to inconsistency. Currently available maps contain significantly more levels of detail (twenty map scales on average) than in the past. These levels must be created, but the optimal strategy to do so is not known. For every user’s data request, a significant part of the data remains the same, but still needs to be included. This leads to more data transfer, and slower response. The interactive Internet environment is not used to its full potential for user navigation. It is common to observe lagging, popping features or flickering of a newly retrieved map scale feature while using the map. This research develops principles of variable scale (vario-scale) maps to address these issues. The vario-scale approach is an alternative for obtaining and maintaining geographical data sets at different map scales. It is based on the specific topological structure called tGAP (topological Generalized Area Partitioning) which addresses the main open issues of current solutions for managing spatial data sets of different scales such as: redundancy data, inconsistency of map scales and dynamic transfer. The objective of this thesis is to design, to develop and to extend the variable-scale data structures and it is expressed as the following research question: How to design and develop a system for vario-scale maps? To address the above research question, this research has been conducted using the following outline:  To address the above research question, this research has been conducted using the following outline: 1) Investigate state-of-the-art in map generalization. 2) Study development of vario-scale structure done so far. 3) Propose techniques for generating better vario-scale map content. 4) Implement strategies to process really massive datasets. 5) Research smooth representation of map features and their impact on user interaction. Results of our research led to new functionality, were addressed in prototype developments and were tested against real world data sets. Throughout this research we have made following main contributions to the design and development of a system of vario-scale maps. We have: studied vario-scale development in the past and we have identified the most urgent needs of the research. designed the concept of granularity and presented our strategy where changes in map content should be as small and as gradual as possible (e. g. use groups, maintain road network, support line feature representation). introduced line features in the solution and presented a fully-automated generalization process that preserves a road network features throughout all scales. proposed an approach to create a vario-scale data structure of massive datasets. demonstrated a method to generate an explicit 3D representation from the structure which can provide smoother user experience. developed a software prototype where a 3D vario-scale dataset can be used to its full potential. conducted initial usability test. All aspects together with already developed functionality provide a more complex and more unified solution for vario-scale mapping. Based on our research, design and development of a system for vario-scale maps should be clearer now. In addition, it is easier to identified necessary steps which need to be taken towards an optimal solution. Our recommendations for future work are: One of the contributions has been an integration of the road features in the structure and their automated generalization throughout the process. Integrating more map features besides roads deserve attention. We have investigated how to deal with massive datasets which do not fit in the main memory of the computer. Our experiences consisted of dataset of one province or state with records in order of millions. To verify our findings, it will be interesting to process even bigger dataset with records in order of billions (a whole continent). We have introduced representation where map content changes as gradually as possible. It is based on process where: 1) explicit 3D geometry from the structure is generated. 2) A slice of the geometry is calculated. 3) Final maps based on the slice is constructed. Investigation of how to integrate this in a server-client pipeline on the Internet is another point of further research. Our research focus has been mainly on one specific aspect of the concept at a time. Now all aspects may be brought together where integration, tuning and orchestration play an important role is another interesting research that desire attention. Carry out more user testing including; 1) maps of sufficient cartographic quality, 2) a large testing region, and 3) the finest version of visualization prototype

Design and development of a system for vario-scale maps

Nowadays, there are many geo-information data sources available such as maps on the Internet, in-car navigation devices and mobile apps. All datasets used in these applications are the same in principle, and face the same issues, namely: Maps of different scales are stored separately. With many separate fixed levels, a lot of information is the same, but still needs to be included, which leads to duplication. With many redundant data throughout the scales, features are represented again and again, which may lead to inconsistency. Currently available maps contain significantly more levels of detail (twenty map scales on average) than in the past. These levels must be created, but the optimal strategy to do so is not known. For every user’s data request, a significant part of the data remains the same, but still needs to be included. This leads to more data transfer, and slower response. The interactive Internet environment is not used to its full potential for user navigation. It is common to observe lagging, popping features or flickering of a newly retrieved map scale feature while using the map. This research develops principles of variable scale (vario-scale) maps to address these issues. The vario-scale approach is an alternative for obtaining and maintaining geographical data sets at different map scales. It is based on the specific topological structure called tGAP (topological Generalized Area Partitioning) which addresses the main open issues of current solutions for managing spatial data sets of different scales such as: redundancy data, inconsistency of map scales and dynamic transfer. The objective of this thesis is to design, to develop and to extend the variable-scale data structures and it is expressed as the following research question: How to design and develop a system for vario-scale maps?  To address the above research question, this research has been conducted using the following outline: 1) Investigate state-of-the-art in map generalization. 2) Study development of vario-scale structure done so far. 3) Propose techniques for generating better vario-scale map content. 4) Implement strategies to process really massive datasets. 5) Research smooth representation of map features and their impact on user interaction. Results of our research led to new functionality, were addressed in prototype developments and were tested against real world data sets. Throughout this research we have made following main contributions to the design and development of a system of vario-scale maps. We have: studied vario-scale development in the past and we have identified the most urgent needs of the research. designed the concept of granularity and presented our strategy where changes in map content should be as small and as gradual as possible (e. g. use groups, maintain road network, support line feature representation). introduced line features in the solution and presented a fully-automated generalization process that preserves a road network features throughout all scales. proposed an approach to create a vario-scale data structure of massive datasets. demonstrated a method to generate an explicit 3D representation from the structure which can provide smoother user experience. developed a software prototype where a 3D vario-scale dataset can be used to its full potential. conducted initial usability test. All aspects together with already developed functionality provide a more complex and more unified solution for vario-scale mapping. Based on our research, design and development of a system for vario-scale maps should be clearer now. In addition, it is easier to identified necessary steps which need to be taken towards an optimal solution. Our recommendations for future work are: One of the contributions has been an integration of the road features in the structure and their automated generalization throughout the process. Integrating more map features besides roads deserve attention. We have investigated how to deal with massive datasets which do not fit in the main memory of the computer. Our experiences consisted of dataset of one province or state with records in order of millions. To verify our findings, it will be interesting to process even bigger dataset with records in order of billions (a whole continent). We have introduced representation where map content changes as gradually as possible. It is based on process where: 1) explicit 3D geometry from the structure is generated. 2) A slice of the geometry is calculated. 3) Final maps based on the slice is constructed. Investigation of how to integrate this in a server-client pipeline on the Internet is another point of further research. Our research focus has been mainly on one specific aspect of the concept at a time. Now all aspects may be brought together where integration, tuning and orchestration play an important role is another interesting research that desire attention. Carry out more user testing including; 1) maps of sufficient cartographic quality, 2) a large testing region, and 3) the finest version of visualization prototype. &nbsp

Data-Driven Estimation in Equilibrium Using Inverse Optimization

Equilibrium modeling is common in a variety of fields such as game theory and transportation science. The inputs for these models, however, are often difficult to estimate, while their outputs, i.e., the equilibria they are meant to describe, are often directly observable. By combining ideas from inverse optimization with the theory of variational inequalities, we develop an efficient, data-driven technique for estimating the parameters of these models from observed equilibria. We use this technique to estimate the utility functions of players in a game from their observed actions and to estimate the congestion function on a road network from traffic count data. A distinguishing feature of our approach is that it supports both parametric and \emph{nonparametric} estimation by leveraging ideas from statistical learning (kernel methods and regularization operators). In computational experiments involving Nash and Wardrop equilibria in a nonparametric setting, we find that a) we effectively estimate the unknown demand or congestion function, respectively, and b) our proposed regularization technique substantially improves the out-of-sample performance of our estimators.Comment: 36 pages, 5 figures Additional theorems for generalization guarantees and statistical analysis adde

Continuum Equilibria and Global Optimization for Routing in Dense Static Ad Hoc Networks

We consider massively dense ad hoc networks and study their continuum limits as the node density increases and as the graph providing the available routes becomes a continuous area with location and congestion dependent costs. We study both the global optimal solution as well as the non-cooperative routing problem among a large population of users where each user seeks a path from its origin to its destination so as to minimize its individual cost. Finally, we seek for a (continuum version of the) Wardrop equilibrium. We first show how to derive meaningful cost models as a function of the scaling properties of the capacity of the network and of the density of nodes. We present various solution methodologies for the problem: (1) the viscosity solution of the Hamilton-Jacobi-Bellman equation, for the global optimization problem, (2) a method based on Green's Theorem for the least cost problem of an individual, and (3) a solution of the Wardrop equilibrium problem using a transformation into an equivalent global optimization problem

State of the art in automated map generalization

Automated map generalization is a difficult, complex and computational very intensive problem. The aim of this chapter is to study existing solutions and state of the art. It also provides context and motivation for why we tackle this problem by applying varioscale approach. In Section 2.1, the paradigm shift in map generalization in a digital environment is studied. We investigate if requirements in the map making process have changed with the transformation from paper to digital environment and if so what are the consequences. Then Section 2.2 investigates how National Mapping Agencies are dealing with automated generalization process in general and what are their recent developments. In Section 2.3, the focus is on the issue of continuous map generalization, which is becoming more researched as an alternative to the map generalization for discrete predefined scales. Section 2.4 demonstrates another problem of digital map environment where the number of map scales available is not sufficient for user interactions. Final remarks are covered in 2.5